30 The Mixing Parameters

In this chapter we review the determination of the mixing parameters q/p for the various neutral-meson systems. We need them because they enter in the rephasing-invariant quantities $λf$ required for the calculation of the CP-violating asymmetries. We identify the conclusions which follow directly and (almost) model-independently from experiment, as opposed to expressions characteristic of the standard model (SM). Extensive use is made of the formulae in Chapter 6.

The analysis presented in this chapter is based on a very simple observation about eqn (6.58),

That observation is the following. If $∣M12∣≫∣Γ12∣$⁠, or else if $ϖ≡arg(M12∗Γ12)$ is very close either to 0 or to $π$⁠, then we can approximate eqn (30.1) by

This equation uses our choice of $Δm>0$ for all neutral-meson systems. On the other hand, if $∣M12∣≪∣Γ12∣$⁠, or else if $ϖ$ is very close either to 0 or to $π$⁠, then we can approximate eqn (30.1) by

If either of these conditions hold, then q/p is a pure phase, which is determined either by the phase of $M12$ or by that of $Γ12$⁠, respectively. The validity of these approximations can be tested either experimentally—see § 30.1.1—or using theoretical arguments—see § 30.1.2.

The magnitude $∣q/p∣=(1−δ)/(1+δ)$ is a function of $δ$⁠. This quantity has only been measured for the neutral-kaon system. For the $Bd0−Bd0¯$ system there is a direct bound on $δ$⁠, while for the $Bs0−Bs0¯$ system that bound arises indirectly, through the bound on the mixing parameter $χ$⁠.

The current experimental bounds may also be used in order to place constraints on the deviation of the argument of q/p from that of $M12∗$⁠, without recourse to theoretical assumptions. One starts from eqn (6.71):

which is exact, and where we have used our convention that $Δm$ is positive. It follows that

Analogously, we may use eqn (6.70),

to show that

The bounds in eqns (30.5) and (30.7) are very useful in the case of the neutral-kaon system. For the B systems they are not that good. There, it is customary to use instead the following theoretical argument.

Suppose that one has an upper bound on the quantity

This quantity is real and non-negative by definition. Let us assume that one can find some argument to show that it is small, $t≪1$⁠. Now, it follows from eqn (30.1) that

From this equation one easily derives

Thus, if we know that t is very small, we learn that q/p has modulus very close to 1, and, therefore, that it is almost a pure phase. From eqn (30.9) one may also derive

Thus, if we know that t is very small, we learn that the phase of q/p is very close to that of $M12∗$⁠.

It is interesting to observe that, while the deviation of $∣q/p∣$ from unity is at most $∼t$⁠, the deviation of arg(q/p) from $argM12∗$ is $∼t2$⁠, and therefore much smaller. This is readily checked by a glance at eqn (30.9): both $tcosϖ$ and $tsinϖ$ must be non-zero in order that $(1−itcosϖ−tsinϖ)(1−itcosϖ+tsinϖ)$ be non-real.

We shall assume that the decay amplitudes corresponding to the dominant decay channels are determined by the SM tree-level diagrams. If there is new physics, then it should affect primarily the mixing in the neutral-meson systems and/or the decays which are suppressed in the SM either by CKM factors, by loop factors, or otherwise. This assumption is useful in order to get an estimate of

The assumption will be used, in the $Bd0−Bd0¯$ and $Bs0−Bs0¯$ systems, to place limits on $∣Γ12∣$⁠, and thus on t. Equations (30.10 and (30.11) are then used to determine q/p through the argument of $M12$⁠.

In the $K0−K0¯$ system all the necessary ingredients are experimentally known and model-independent statements are possible. The determination of q/p follows most easily from eqn (30.6). Remember that in the derivation of this equation, which is exact, we have used our convention that $Δm$ is positive. Experimentally, $u≡−ΔΓ/(2Δm)=1.054±0.004$⁠, while $δ=(3.27±0.12)×10−3$⁠. It follows that

and therefore q/p is almost a pure phase, which is given by

In the calculation of $argΓ12∗$ one first uses the result (Lavoura 1992a)⁷⁹

to trade the phase of $Γ12∗$ for that of $A0A¯0∗$⁠. One then assumes that, even if there is physics beyond the SM, the $∣ΔS∣=1$ decay amplitudes are dominated by the usual charged-current weak interaction. Then, as in eqn (17.5),

One concludes that

This result hinges on experimental facts and on the assumption that the kaon decays to $2π,I=0$ are dominated by the tree-level diagram with an intermediate W boson. Equation (30.17) neglects a term $δ∼10−3$ in the phase of $qK/pK$ (see eqn 30.14).

Instead of eqn (30.17), one may write

making use of the phase $ϵ′≡arg(−VusVcdVud∗Vcs∗)$ introduced by Aleksan et al. (1994). The equality between the right-hand sides of eqns (30.17) and (30.18) is exact: it holds in any model, because it follows from the definition of $ϵ′$ (Cohen et al. 1997).

In the SM, $ϵ′∼10−3$ may be safely neglected. In most models beyond the SM $ϵ′$ is still very small—see § 28.5.2. Moreover, models in which $ϵ′$ is altered usually produce a large effect on $B0−B0¯$ mixing—see for instance Nir and Silverman (1990). For this reason, many authors set $ϵ′=0$ in eqn (30.18). Indeed, if $ϵ′∼10−3$⁠, then neglecting $ϵ′$ is just as good an approximation as neglecting $δ$⁠, as we did when writing down both eqns (30.17) and (30.18). On the other hand, if in a model beyond the SM $ϵ′$ turns out to be large, then it makes sense (Kurimoto and Tomita 1997) to use eqn (30.18) while neglecting $δ$⁠, because the latter quantity is experimentally small. This is why we prefer to keep $ϵ′$ explicit in our formulae.

Instead of starting from eqn (30.6) as we did, we might start from eqn (30.4). As $δ$ is small and $u≈1$ one may write

Comparing eqns (30.14) and (30.19) we see that $ϖ≡arg(M12∗Γ12)≈π−2δ$⁠, cf. eqn (8.15).

At this juncture, we should stress the important constraints placed by the experimental value of $δ$ on model-building. We have discussed the standard-model computation of $M12∗$ in Chapter 17. The short-distance contribution comes from the box diagram. If we omit the spurious phase $ξK+ξd−ξs$⁠, that contribution may be written as the sum of three pieces, proportional to $λu2,λuλc,$ and $λc2$⁠, respectively, where $λα=Vαs∗Vαd$⁠. There is also a long-distance contribution to $M12∗$⁠; its phase is that of $λu2$⁠.

Thus, the calculation of $M12∗$ involves three terms with different phases. By contrast, the calculation of $−Γ12∗$ involves a single weak phase, that of $λu2$⁠. But the phases of $M12∗$ and of $−Γ12∗$ have to coincide up to corrections $∝δ∼10−3$⁠. In the SM this constraint is satisfied, because the phases of $λu$ and of $−λc$ coincide up to $ϵ′∼10−3$⁠.

The same argument should hold if there is physics beyond the SM. Let us consider a model with no new contributions to $Γ12∗$⁠, but with an effective $∣ΔS∣=2$ contribution to $M12∗$⁠. It is an experimental requirement that this extra contribution to $M12∗$ should retain a phase close to that of $−Γ12∗$⁠. One also has to be careful that the phases of $−λc$ and of $λu$ do not get too different from each other; otherwise the box diagram produces $M12∗$ with a phase very different from that of $−Γ12∗$—barring large cancellations between the box diagram and the new-physics contributions to $M12∗.$ This is another argument for the smallness of $ϵ′$ even when there is physics beyond the SM.

The CLEO Collaboration (1993c) has established that $∣δ∣<0.09$ in the $Bd0−Bd0¯$ system. We also know that $Δm$ must be greater than $∣ΔΓ∣$⁠; this arises from the combination of the observed oscillation in time-dependent measurements, with the time-independent determination of $χ$⁠, as discussed in Chapter 10. The Particle Data Group (1996) finds $x=0.73±0.05$⁠, while, based on the data, one may conclude that $∣y∣<0.3$ at 95% confidence level (Moser, personal communication). It follows that $∣u∣=∣y/x∣<0.47$⁠, and $∣uδ∣<0.043$⁠. From eqn (30.4) one then knows that q/p is almost a pure phase, and, using eqn (30.5), we find that that phase does not deviate from that of $M12∗$ by more that $2.5°$⁠.

One may do much better than this by using the theoretical argument in § 30.1.2. For this we need a bound on t. The experimental bound is rather poor. Indeed, as $∣δ∣$ is very small and $∣ΔΓ∣<Δm$⁠, one learns from eqn (6.64) that $∣M12∣≈Δm/2$⁠. Combining this with $∣Γ12∣≤Γ$ we find the constraint

This is rather useless.

But, let us assume that the contributions to $Γ12$ from decay channels common to both $Bd0$ and $Bd0¯$ do not differ much from the standard-model expectations. This is likely to be the case, since these decays should be dominated by SM tree-level diagrams (Nir 1993). The branching ratios to channels common to $Bd0$ and $Bd0¯$ are expected to be no larger than $10−3$ in the SM. Moreover, in the determination of $Γ12$ via eqn (30.12) these contributions should appear with different signs. Hence, even if we allow for an enhancement by a factor of 10 due to the adding up of different contributions, one finds $∣Γ12∣/Γ<10−2$ (Nir 1993). Using this result instead of $∣Γ12∣≤Γ$ in eqn (30.20), one obtains

Thus, $t<0.015$⁠. This guarantees that the phase of q/p deviates from that of $M12∗$ by no more than $0.007°$⁠, which is a much better result than those in the previous paragraphs. In addition, using eqn (6.28), we also find that $∣δ∣≤t≤0.015$⁠.

In the $Bs0−Bs0¯$ system we take $x≥9.5$ and $∣y∣≤0.3$⁠, cf. Chapter 10. Then, using eqn (6.54),

we find

while

Therefore, $∣δu∣<0.003$⁠, and the phase of q/p deviates from that of $M12∗$ by no more than $0.18°$⁠.

As $∣δ∣$ is small and $∣y∣≪x$ one may, just as in the $Bd0−Bd0¯$ system, write $∣M12∣=Δm/2$⁠. We then find

As we did in § 30.3.1, we may find a better bound on t if we assume that the decay amplitudes to channels common to $Bs0$ and $Bs0¯$ do not differ much from the standard-model expectations. The dominant decays common to $Bs0$ and $Bs0¯$ are due to the tree-level transitions $b→cc¯s$⁠; all other decays are CKM-suppressed relative to this one and should play a minor role. Therefore, $Γ12/Γ$ is expected to be large. Indeed, Beneke et al. (1996) found, in the SM,

in a convention in which no phases are introduced by the CP transformations.⁸⁰ Since the analysis involves primarily tree-level decays, it is likely to hold in many models beyond the SM. Using the highest value in eqn (30.26), $∣Γ12∣∼0.14Γ$⁠, one obtains

By coincidence, one arrives at the same final estimate as for the $Bd0−Bd0¯$ system, cf. eqn (30.21). In the $Bs0−Bs0¯$ system the bound on $∣Γ12∣/Γ$ is worse, but that is compensated by the larger value of x. Therefore, the phase of q/p deviates from that of $M12∗$ by no more than $0.007°$⁠, improving upon the experimental result. Again, using eqn (6.28), we find that $∣δ∣≤t≤0.015$⁠.

We have shown in the previous section, using only experimental results, that

holds up to a reasonable accuracy. We have then followed Nir (1993) and used a mild assumption about the decay amplitudes to argue that eqn (30.28) should, in fact, be valid to extremely good accuracy. We must now determine the phase of $M12∗$⁠. This is readily done in the SM.

In the SM, an analysis of the box diagram yields

thus vindicating eqn (30.28). Using the determination of $M12$ in eqn (18.1), we find

The CKM combination $Vtb∗Vtq$ appears because, as explained in Chapter 18, the SM box diagram for $M12$ is dominated by the contribution with two internal top quarks. Long-distance contributions to $M12$ should be negligible. The ‘bag parameter’ $BBq$ is expected to be positive. However, if the vacuum-insertion approximation fails badly, $BBq$ might be negative (Grossman et al. 1997b) in either the $Bd0−Bd0¯$ system or the $Bs0−Bs0¯$ system, or in both of them.

New physics may change the phase of $qBq/pBq$⁠, or even lead to $∣qBq/pBq∣≠1$⁠. In some models, new physics appears predominantly in the mixing of the neutral mesons, while their decays remain governed by SM physics—examples are given by Nir and Silverman (1990), Dib et al. (1991), and Branco et al. (1993). This is especially likely when the decays occur at tree level in the SM. Then, the effects of new physics on CP asymmetries may be parametrized by parameters r and $θ$ as

It follows that

(We take r real and positive by definition. Of course, r and $θ$ are in principle different in the $Bd0−Bd0¯$ and $Bs0−Bs0¯$ systems.) We thus have

This treatment of physics beyond the SM is originally due to Soares and Wolfenstein (1993) and to Branco et al. (1993), and has also been used by Deshpande etal. (1996), Cohen et al. (1997), Grossman et al. (1997a), and Silva and Wolfenstein (1997).⁸¹

We have argued in this chapter that q/p is given, to an excellent approximation, by a pure phase. In the kaon system this is a direct result of the experimental determination of $δ$⁠. In the $Bd$ and $Bs$ systems, we may use the limited experimental results to ascertain that the phase of q/p differs from the pure phase $M12∗/∣M12∣$ by no more than $2.5°$ and $0.2°$⁠, respectively. Alternatively, we may follow Nir (1993) and assume that the dominant contributions to $∣Γ12∣$ arise from SM diagrams. We then conclude that the deviation is at most $0.007°$⁠, in both the $Bd$ and $Bs$ systems.

In the case of $qK/pK$⁠, we have argued that its phase should retain the expression that it has in the SM, even in the presence of new physics effects. On the other hand, the phases $qBq/pBq$ could differ significantly from their SM values. We parametrize this difference by phases $θq$⁠, which arise from possible new-physics effects.

We found

In each equation, the second line contains in between squared brackets the SM result, with the phases in the CKM matrix parametrized following the convention of eqn (28.24). We see that, if the mixing is only due to SM diagrams, then q/p may be parametrized exclusively in terms of $β,ϵ,$ and $ϵ′$⁠, even if the CKM matrix is not unitary.